Extremely closed subgroups and a variant on Glauberman's $Z^*$-theorem

Hung P. Tong-Viet

Vol. 329 (2024), No. 2, 303–325

DOI: 10.2140/pjm.2024.329.303

Abstract

Let $G$ be a finite group and let $H$ be a subgroup of $G$. We say that $H$ is extremely closed in $G$ if $⟨H,H^g⟩\cap N_G(H)=H$ for all $g\in G$. We determine the structure of finite groups with an extremely closed abelian $p$-subgroup for some prime $p$. In particular, we show that if $G$ contains such a subgroup $H$, then $G=N_G(H)O_{p^{\prime }}(G)$. This is a variant on the celebrated Glauberman’s $Z^{\ast }$-theorem.

Keywords

extremely closed, strongly closed, weakly closed

Mathematical Subject Classification

Primary: 20D05, 20D15

Milestones

Received: 15 April 2024

Revised: 10 May 2024

Accepted: 23 June 2024

Published: 6 July 2024